How to Find the Radius of a Cylinder from Its Volume

To find the radius of a cylinder from its volume, use the formula r = √(V / (π × h)) — divide the volume by Pi times the height, then take the square root. For example, a cylinder with a volume of 785.40 cm³ and a height of 10 cm has a radius of √(785.40 / (3.14159 × 10)) = √25 = 5 cm.

This reverse cylinder volume formula works for any solid right cylinder where you know both the volume and the height. In this guide, you'll get the full derivation, 5 worked examples in metric and imperial units, real-world applications (water tanks, engine bores, pipes), and a handy reference table. You can verify every answer instantly using the free cylinder volume calculator on our homepage.

Quick Answer: The Radius of a Cylinder Formula

Formula: r = √(V / (π × h))

Best for: Students solving geometry problems, engineers sizing hydraulic cylinders, DIY homeowners designing water tanks, and anyone who knows the volume and height of a cylinder but needs to find its radius.

What Is the Radius of a Cylinder?

The radius of a cylinder is the distance from the center of its circular base to the edge of that base. Because a right cylinder has two identical circular bases, both share the same radius.

The radius is exactly half the diameter and connects directly to every other cylinder measurement — base area (πr²), lateral surface area (2πrh), and total volume (πr²h). Knowing the radius lets you calculate any other property of the cylinder. You can explore the full anatomy of the shape in our guide on the parts of a cylinder.

Deriving the Radius Formula from V = πr²h

The standard cylinder volume formula explained is:

V = π × r² × h

To solve for r, rearrange the equation step by step:

1. Divide both sides by π × h: r² = V / (π × h)
2. Take the square root of both sides: r = √(V / (π × h))

That's it. The formula tells you that the radius depends on two things — the cylinder's volume and its height. If you only know the volume but not the height, the radius cannot be determined, because infinitely many cylinders can have the same volume (tall and thin, or short and wide).

This is a direct consequence of the base area principle from cylinder geometry: volume equals base area times height, so dividing volume by height recovers the base area (πr²), from which the radius follows. For a deeper mathematical treatment, see cylinder (MathWorld).

How to Find the Radius of a Cylinder: Step-by-Step

Follow these 4 steps every time:

1. Check your units match. Volume and height must use compatible units. If volume is in cm³, height must be in cm. If volume is in cubic inches (in³), height must be in inches.
2. Multiply the height by π. Use π ≈ 3.14159 for accurate results.
3. Divide the volume by that result. This gives you r² (the squared radius).
4. Take the square root. The result is the radius in the same linear units as the height.

Worked Example

• Volume = 1,000 cm³, Height = 10 cm
• Step 1: Units match (cm³ and cm) ✅
• Step 2: π × 10 = 31.4159
• Step 3: 1,000 ÷ 31.4159 = 31.831
• Step 4: √31.831 = 5.64 cm

You can verify this instantly using our free cylinder volume calculator — enter 5.64 cm as the radius and 10 cm as the height, and it will return 1,000 cm³.

5 Worked Examples (Metric & Imperial)

Example 1 — Volume in Cubic Centimeters

Given: V = 500 cm³, h = 8 cm

• r² = 500 / (3.14159 × 8) = 500 / 25.1327 = 19.89
• r = √19.89 = 4.46 cm

Example 2 — Volume in Cubic Inches

Given: V = 200 in³, h = 12 in

• r² = 200 / (3.14159 × 12) = 200 / 37.699 = 5.305
• r = √5.305 = 2.30 inches

Need this in another imperial unit? Try the dedicated calculator in inches.

Example 3 — Water Tank (Volume in Liters)

A cylindrical water tank holds 250 liters and is 1.2 meters tall. What's its radius?

First, convert liters to cubic meters (1,000 L = 1 m³):

• 250 L = 0.25 m³
• r² = 0.25 / (3.14159 × 1.2) = 0.25 / 3.770 = 0.0663
• r = √0.0663 = 0.2575 m = 25.75 cm

So the tank has a radius of about 25.75 cm (diameter of 51.5 cm). For similar tank-sizing problems, use the cylinder tank calculator.

Example 4 — Volume in US Gallons

Given: V = 50 US gallons, h = 3 feet

Convert gallons to cubic feet (1 ft³ = 7.4805 US gal):

• 50 ÷ 7.4805 = 6.684 ft³
• r² = 6.684 / (3.14159 × 3) = 6.684 / 9.4248 = 0.709
• r = √0.709 = 0.842 ft ≈ 10.11 inches

For direct conversions, see the gallons to liters converter.

Example 5 — Engine Cylinder Bore

An engine cylinder has a swept volume (displacement) of 500 cm³ per cylinder and a stroke (height) of 86 mm (8.6 cm). Find the bore radius:

• r² = 500 / (3.14159 × 8.6) = 500 / 27.017 = 18.507
• r = √18.507 = 4.30 cm

So the bore diameter is 8.6 cm (86 mm) — a typical passenger car dimension. Engineers can cross-check this with our engine cylinder volume calculator.

Finding the Radius When Volume Is in Liters or Gallons

Because the formula r = √(V / (π × h)) requires cubic units that match your height units, always convert liquid-volume units to cubic units first:

For volumes already in cubic units (cm³, m³, in³, ft³), skip the conversion and go straight to the formula. Learn more about standard conversions from standard units (NIST).

Alternative Ways to Find the Radius

Sometimes you don't have the volume. Here are three alternate methods.

From the Diameter

If you know the diameter (d), the radius is simply half of it:

r = d / 2

For a cylinder with d = 12 cm, r = 6 cm. This is the fastest route and avoids any square roots. See our cylinder volume using diameter calculator for full volume calculations.

From the Circumference

If you know the circumference (C) of the circular base:

r = C / (2π)

For C = 31.42 cm, r = 31.42 / 6.2832 = 5 cm. This is useful when you can wrap a measuring tape around a pipe or tank but can't measure inside. Try the calculate from circumference tool.

From the Surface Area

If you know the total surface area (A) and the height:

r = (−h + √(h² + 2A/π)) / 2

This quadratic form is more complex but useful in engineering when surface area is known from material costs or coatings.

Real-World Applications

Finding the radius from volume comes up more often than most people think:

• Water storage tanks: Designers know the required capacity (in liters or gallons) and the available vertical space; they solve for radius to size the tank correctly.
• Pipe sizing: Plumbers calculate required internal radius when flow volume per second and pipe length are known.
• Engine design: Automotive engineers work backward from target displacement and stroke length to find the bore radius.
• Chemistry labs: Technicians design reaction vessels to hold exact liquid volumes at a fixed height.
• Packaging: Product designers fit a required liquid volume into a container of fixed height (e.g., a 330 mL soda can at 115 mm tall).
• Silos and grain storage: Farmers compute the required silo radius from desired capacity and the maximum practical height.

Every one of these applications uses the same formula r = √(V / (π × h)) — only the units and scale change.

Common Mistakes to Avoid

Watch for these pitfalls, because they silently corrupt every calculation downstream:

• Mismatched units. If volume is in cm³, height must be in cm. Using cm³ with height in meters produces a wrong answer by a factor of 100.
• Forgetting to take the square root. Stopping at V / (πh) gives r², not r. Always finish with √.
• Using π = 3 or π = 22/7 for precision work. For 2-decimal accuracy, use π ≈ 3.14159.
• Confusing diameter with radius. Some problems give the diameter of the base; halve it before applying the volume formula.
• Using slant height instead of perpendicular height. For an oblique cylinder calculator, h must be the perpendicular distance between bases, not the slant.
• Rounding too early. Round only the final answer. Rounding intermediate steps can shift the radius by 1–2%.
• Applying this formula to hollow cylinders. Hollow cylinders need a different approach — try the hollow cylinder calculator.

Radius Formulas Quick Reference Table

Frequently Asked Questions

Can you find the radius of a cylinder from volume alone?

No — you need at least one more dimension. The volume formula V = πr²h has two unknowns (r and h), so a single value can correspond to infinitely many cylinders. A tall, thin cylinder and a short, wide cylinder can share the same volume. You must know either the height, diameter, circumference, or surface area in addition to the volume.

What is the formula for the radius of a cylinder?

The formula is r = √(V / (π × h)) when volume and height are known. This comes from rearranging the standard volume formula V = πr²h. Plug in V (volume), π (≈3.14159), and h (height) to find r. Our cylinder volume calculator tool lets you verify any radius by entering it back into V = πr²h.

How do you find the radius of a cylinder in inches?

Use the same formula r = √(V / (π × h)), with V in cubic inches and h in inches. For example, if V = 150 in³ and h = 10 in, r = √(150 / (π × 10)) = √4.77 = 2.18 inches. If your volume is in US gallons, first convert: 1 gal = 231 in³. For measurements already in inches, use our calculator in inches.

How do I find the radius of a cylinder when volume is in liters?

Convert liters to cubic centimeters (× 1,000) or cubic meters (÷ 1,000), then apply r = √(V / (π × h)). Example: A 2-liter cylinder 20 cm tall: V = 2,000 cm³, r = √(2,000 / (3.14159 × 20)) = √31.83 = 5.64 cm. Always match the volume units to the height units. The volume in liters page has a calculator that handles this automatically.

Why do I need the height to find the radius?

Because volume is a product of two independent dimensions — base area and height. Knowing only the volume is like knowing only the area of a rectangle: you can't tell whether it's a 2×6 or a 3×4 without one more piece of information. In the cylinder case, the height "anchors" the equation so the radius becomes solvable.

Is the formula different for a hollow cylinder?

Yes — hollow cylinders require solving for either the outer radius (R) or the inner radius (r), given V = π × h × (R² − r²). You need the volume, height, and one of the two radii to find the other. The formula becomes R = √((V / (π × h)) + r²). For these calculations, use our dedicated hollow cylinder calculator.

How accurate is π = 3.14 versus π = 3.14159?

Using π = 3.14 introduces an error of about 0.05% in the radius. For classroom work, this is fine. For engineering tolerances (e.g., engine bore specs), use at least π ≈ 3.14159 — this gives accuracy to 4-5 decimal places. For high-precision scientific applications, most calculators store π to 15 digits (3.141592653589793).

How do you find the diameter once you know the radius?

Multiply the radius by 2: d = 2r. If you've calculated r = 4.46 cm, the diameter is 8.92 cm. This is often the more practically useful number because diameter is easier to measure with calipers or rulers on a physical cylinder. Educational resources like Khan Academy geometry explain this relationship in depth.

Can this method be used for oblique or slanted cylinders?

Yes, as long as h is the perpendicular height (the shortest distance between the two parallel bases), not the slant length. By Cavalieri's principle, an oblique cylinder has the same volume as a right cylinder with the same radius and perpendicular height. So the radius formula r = √(V / (π × h)) still applies.